Answer:
[tex]x = \bold{0.414} \textrm{ and } x= \bold{-2.414}[/tex]
Step-by-step explanation:
See attached graph for a visual
The x-intercepts of a graph are where the graph crosses the x-axis ie where the value of y = 0 The equation of the graph is y=-x^{2}-2x+1
Setting y = 0 and solving for the resultant equation will provide the x-intercepts
[tex]0=-x^{2}-2x+1or-x^{2}-2x+1=0[/tex]
Reversing the signs gives us
[tex]x^{2}+2x-1=0[/tex]
This is a quadratic equation of the form [tex]ax^{2}+bx+c=0[/tex] with a = 1, b = 2 and c=-1
For an equation of this form the roots of the equation ie the values of x which satisfy the equation are given by
[tex]\frac{-b\pm\sqrt{{b^{2}-4ac}}}{2a}[/tex]
Substituting we get the two roots as
[tex]x=\frac{-2\pm\sqrt{{2^{2}-4(1)(-1}}}{2}=\frac{-2\pm\sqrt{{4-(-4)}}}{2}x=\frac{-2\pm\sqrt{8}}{2}8 = 4(2)\sqrt{8}=\sqrt{4}\sqrt{2} = 2\sqrt{2}[/tex]
So the two roots are
[tex]x=\frac{-2\pm2\sqrt{2}}{2}[/tex]
Dividing by 2 we get the two roots(x-intercepts) as
[tex]x=-1\pm\sqrt{2}\\\\x_{1}=-1+\sqrt{2} =-1+1.414=\bold{0.414}[/tex]
[tex]x_{2}=-1-\sqrt{2}=-1-1.414=\boldsymbol{-2.414}[/tex]
